Formally, \(P_0(\mathbf {x}) \overset{d}{=}P_1(\mathbf {x}), P_0(y \mid \mathbf {x}) \overset{d}{=}P_1(y \mid \mathbf {x})\) .\(\overset{d}{=}\) : Equal in distribution. This means that \(A\)  has no effect on \(X\)  or \(Y\) . In this setting, both population subgroups have the same distribution of features/covariates, and the same probability of disease given specific covariate values. Note that \(A\)  is independent of \(X\) : not only is \(A\)  not included in \(X\) , but no component of \(X\)  is affected by \(A\) . This is an idealized setting in which there are no differences between subgroups in (1) the distribution of covariates and (2) their relationship to the outcome of interest. However, this is unlikely in practice: even when \(A\)  is not used as a model input, proxies for \(A\)  can appear in \(X\)  such that \(P_0(\mathbf {x}) \overset{d}{\ne }P_1(\mathbf {x})\) , as discussed in [1]}, [2]}.\(\overset{d}{\ne }\) : not equal in distribution. We provide examples of such proxies in the next setting.
